\documentclass{seminar}

\usepackage{amssymb,amsmath, amscd, enumerate}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{maintheorem}[theorem]{Main  Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{definition}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{question}[theorem]{Question}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newcommand{\f}[1]{\ensuremath{\mathfrak{#1}}}
\newcommand{\ol}{\overline}
\DeclareMathOperator{\Deg}{deg} \DeclareMathOperator{\Ht}{ht}
\DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\reg}{reg} \DeclareMathOperator{\en}{end}
\DeclareMathOperator{\Rad}{Rad} \DeclareMathOperator{\Ass}{Ass}
\DeclareMathOperator{\Spf}{Spf} \DeclareMathOperator{\hgt}{ht}
\DeclareMathOperator{\ord}{ord}


\DeclareMathOperator{\Gr}{gr} \DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\ecodim}{ecodim}
\DeclareMathOperator{\projdim}{projdim}
\DeclareMathOperator{\emdim}{emdim}
\DeclareMathOperator{\grade}{grade}
\DeclareMathOperator{\depth}{depth} \DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\F}{\mathcal F}
\DeclareMathOperator{\Q}{\mathcal Q}

\DeclareMathOperator{\Z}{\mathbb Z}

\DeclareMathOperator{\N}{\mathbb N}


\DeclareMathOperator{\q}{\bf q} \DeclareMathOperator{\m}{\bf m}
\DeclareMathOperator{\n}{\bf n}


\def\demo{\noindent{\bf Proof. }}
\def\sqr#1#2{{\vcenter{\hrule height.#2pt
        \hbox{\vrule width.#2pt height#1pt \kern#1pt
                \vrule width.#2pt}
        \hrule height.#2pt}}}
\def\square{\mathchoice\sqr64\sqr64\sqr{4}3\sqr{3}3}
\def\QED{\hfill$\square$}

\def\tratto{\mbox{\rule{2mm}{.2mm}$\;\!$}}


\newcommand{\heading}[1]{\begin{center}\small\bf #1\end{center}}
\begin{document}

\begin{slide}
\heading{GENERIC FIBER  RINGS OF MIXED }

\heading{ POLYNOMIAL/POWER SERIES RINGS }

\vspace{0.2in}


\begin{center}
William Heinzer
\end{center}

\begin{center}
Purdue University
\end{center}

\vspace{0.2in}

\begin{center}
joint with Christel Rotthaus and Sylvia Wiegand
\end{center}
\end{slide}




\begin{slide}

\heading{MOTIVATION}

A question raised by Mel Hochster:

\medskip

{\bf Question.}  Can one describe or somehow classify the local
maps $R \hookrightarrow S$ of complete local domains  $R$ and $S$
such that $$  P\in\Spec(S), \quad P\ne (0)  \implies P\cap R\ne
(0)?$$

\medskip

Hochster remarks  that if,  for example, $R$  is equal
characteristic  zero, such extensions arise as
$$
R = K[[x_1,\dots,x_m]] \hookrightarrow T =
L[[x_1,\dots,x_m,y_1,\dots,y_n]] \to T/P = S,
$$
where $K$ is a subfield of $L$ and $P\in \Spec(T)$ is maximal with
respect to $P\cap R=\{0\}$.

{\bf Partial Answer.} In Hochster's set-up,

\center{$[L:K]<\infty\implies \dim S=2$ or $\dim S=m$.}

\end{slide}



\begin{slide}
\begin{center}
\textbf{Main Results}
\end{center}
Let $K$ be a field, $m$ and $n$ positive integers, $X = \{x_1,
\ldots, x_m\}$,

$Y = \{y_1, \ldots, y_n\}$ sets of independent variables over $K$.

Let $A$ be the localized polynomial ring $K[X]_{(X)}$.

\medskip

{\bf Theorem 1.} Every prime ideal $P$ of $\widehat A = K[[X]]$
that is maximal with respect to $P \cap A = (0)$ has height $m-1$.

\medskip

Let \quad $B := K[[X]][Y]_{(X, Y)}$ \quad and \quad $C :=
K[Y]_{(Y)}[[X]]$.

\medskip

{\bf Theorem 2.} Every prime ideal $P$ of $\widehat B = \widehat
C$ that is maximal with respect to either $P \cap B = (0)$ or $P
\cap C = (0)$,  has height $m+n-2$.

\medskip

{\bf Theorem 3.} Every  prime ideal   $P$ of $K[[X,Y]]$ that is
maximal with respect to $P \cap K[[X]] = (0)$ has height either
$n$ or $m+n-2$.

\end{slide}


\begin{slide}
\begin{center}
\textbf{Local Embeddings}
\end{center}
There exist local embeddings:

\medskip

$ A = K[X]_{(X)}\hookrightarrow \widehat A:=K[[X]],\qquad \widehat
A \hookrightarrow \widehat B = \widehat C = K[[X, Y]] $

\medskip

$ B = K[[X]]\,[Y]_{(X,Y)}\hookrightarrow C =
K[Y]_{(Y)}[[X]]\hookrightarrow\widehat B=\widehat C=
K[[X]]\,[[Y]]. $

\medskip
Matsumura observes there exist $P\in \Spec\widehat A $ with $ \hgt
P = m-1$

and $ P \cap A = (0)$, and also that there exist $ P\in \Spec
K[[X, Y]]$

 with $\hgt P = m+n-2$ such that

 \center{$ P \cap C = (0)$ \quad or \quad
 $P \cap B = (0)$.}

However he does not address the question of whether all primes

maximal with these properties   have the same height.

\end{slide}



\begin{slide}
\begin{center}
\textbf{Generic Fiber Rings}
\end{center}

Let $R \hookrightarrow S$ be an injective homomorphism of
commutative rings

with  $R$  an integral domain. The {\bf generic fiber ring } of
the map

$R \hookrightarrow S$ is the localization $(R \setminus
(0))^{-1}S$ of $S$.

\medskip

Let $(R,\m)$ be a Noetherian local integral domain and let
$\widehat R$ denote the $\m$-adic completion of $R$.

\medskip

The {\bf generic formal fiber ring} of  $R$ is the localization

$(R \setminus (0))^{-1}\widehat R$ of $\widehat R$.

\medskip

The {\bf formal fibers} of $R$ are the fibers of the map

$\Spec \widehat R \to \Spec R$.    For $P \in \Spec R$, the  {\bf
formal fiber}  over $P$ is

$\Spec(\,(R_P/PR_P)\otimes_R\widehat R$\,).

\end{slide}





\begin{slide}
\begin{center}
\textbf{Trivial Generic Fiber Extensions}
\end{center}

Let $R$ be a subring of an integral domain $S$.

{\bf  Definition.}  $R\hookrightarrow S$ is a {\bf trivial generic
fiber} extension or a

{\bf TGF} extension if

\center{$(0)\ne P\in \Spec S \implies P\cap R\ne (0).$}

\medskip

One obtains a TGF extension $S$ of $R$ by considering
$$
R\hookrightarrow T \to T/P := S,
$$ where $T$ is an
extension ring of $R$  and $P \in \Spec T$ is maximal with

respect to $P\cap R=(0)$.

Thus the generic fiber ring of $R \hookrightarrow T$ is relevant
to constructing

TGF extensions $S$ of $R$.




\end{slide}


\begin{slide}
\begin{center}
\textbf{Variations on a theme of Weierstrass}
\end{center}

The Weierstrass Preparation Theorem is our main technical tool.

Let  $P$ be a prime ideal  in the power series ring $\widehat A =
K[[X]]$, where $X = \{x_1, \ldots , x_n\}$ is a set of $n$
variables over the field $K$.

Here $A = K[X]_{(X)}$ is the localized polynomial ring.

{\bf  Notation.} By a {\bf change of variables}, we  mean a finite
sequence of  `polynomial' change of variables  over $K$ of the
form
$$
x_1 \mapsto x_1 +x_n^{e_1}, \quad \ldots \quad  x_{n-1} \mapsto
x_{n-1} + x_n^{e_{n-1}}, \quad  x_n \mapsto x_n,
$$ perhaps  followed by
$$
z_1 \mapsto z_1,\quad z_2 \mapsto z_2 + z_1^{f_2},\quad \ldots
\quad z_n \mapsto z_n+ z_1^{f_n},
 $$
where $e_i, f_i \in \N$ and $z_i$ denotes the image of $x_i$ under
the first map.

\end{slide}


\begin{slide}
\begin{center}
\textbf{The Weierstrass Preparation Theorem}
\end{center}
{\bf Theorem (Weierstrass)} Let $( R, \m)$ be a complete local
ring, let $f \in R[[x]]$ be a formal power series and let
$\overline f$ denote the image of $f$ in $(R/\m)[[x]]$. Assume
that $\overline f \ne 0$ and that $\ord \overline f = s > 0$.
There exists a unique ordered pair $(u, F)$ such that $u$ is  a
unit in $R[[x]]$ and  $F \in R[x]$ is a distinguished monic
polynomial of degree $s$ such that $f = uF$.

Here $F = x^s + a_{s-1}x^{s-1} + \cdots + a_0 \in R[x]$ is {\bf
distinguished} if $a_i \in \m$ for $0 \le i \le s-1$.

\medskip

{\bf Corollary.} The ideal $fR[[x]]$ is extended from $R[x]$ and
$R[[x]]/(f)$ is a free $R$-module of rank $s$. Every $g \in
R[[x]]$ is of the form $g = qf + r$, where $q \in R[[x]]$ and $r
\in R[x]$ is a polynomial with $\deg r \le s-1$.

\end{slide}


\begin{slide}
\begin{center}
\textbf{Embeddings of Power Series Rings}
\end{center}
{\bf Theorem (Classical)} There exists a $K$-algebra embedding of
the formal power series ring $R :=K[[x_1, \ldots, x_n]]$ into the
formal power series ring in two variables $K[[y, z]]$.

{\bf Proof.} Let $f_1, \ldots, f_n$ be power series in $K[[y]]$
that are algebraically independent over $K$. Define $\varphi : R
\to K[[y, z]]$ by $\varphi(x_i) = f_iz$,  for $1 \le i \le n$.
Suppose $g \in R$ is such that $\varphi(g) = 0$. Write $g =
\bigoplus_{k=0}^\infty  g_k$, where $g_k$ is a form in $K[x_1,
\ldots, x_n]$ of degree $k$. Then $\varphi(g_k) = z^kg_k(f_1,
\ldots, f_n)$ with $g_k(f_1, \ldots, f_n) \in K[[y]]$. Hence
$$
\varphi(g) = \bigoplus_{k=0}^\infty g_k(f_1, \ldots, f_n)z^k = 0
$$
implies $g_k(f_1, \ldots, f_n) = 0$ for each $k \in \N$. Since
$f_1, \ldots, f_n$ are algebraically independent over $K$, each
$g_k = 0$, so $g = 0$.


\end{slide}


\begin{slide}
\begin{center}
\textbf{Cohen's Theorem 8}
\end{center}

{\bf Theorem (Classical)} Let $I$ be an ideal of a ring $R$ and
let $M$

be an $R$-module. Assume that $R$ is complete in the $I$-adic
topology

and $\bigcap_{n=1}^\infty I^nM = (0)$. If $M/I$ is generated over
$R/I$ by elements

$\overline{w}_1, \ldots, \overline{w}_s$ and $w_i$ is a preimage
in $M$ of $\overline{w}_i$ for $1 \le i \le s$, then

$M$ is generated over $R$ by $w_1, \ldots, w_s$.

\medskip

{\bf Corollary.} Assume  $\varphi :(R, \m) \to (S, \n)$ is a local
homomorphism.

If $R$ is $\m$-adically  complete, $\m S$ is $\n$-primary and
$S/\n$ is finite

over $R/\m$, then $S$ is a finitely generated  $R$-module.

If $\varphi : R \to K[[y, z]] = S$ is an embedding of the power
series ring in

$n > 2$ variables, then $\m S$ is not $\n$-primary.


\end{slide}

\begin{slide}

\heading{The Generic Formal Fiber Rings of  $B$ and $C$}

\smallskip

Since  $ B := K[[X]][Y]_{(X, Y)} \hookrightarrow K[Y]_{(Y)}[[X]]
:= C $,

\smallskip

if  $P \in \Spec K[[X, Y]]$ and  $P \cap C = (0)$, then $ P \cap B
= (0)$.

\smallskip

We know  $P\in\Spec K[[X, Y]]$ maximal with respect to $P \cap B =
(0)$

\smallskip

or maximal with respect to $P \cap C = (0)\implies \hgt P=n + m -
2$.

\smallskip

Hence if $P \in \Spec K[[X, Y]]$ is maximal  with  $P \cap C =
(0)$,

\smallskip

then $P$ is also maximal  with respect to $P \cap B = (0)$.

\smallskip

However, if  $n \ge 2$, the generic fiber of $B \hookrightarrow C$
is nonzero.

\smallskip

This means  $\exists \quad  P\in\Spec K[[X, Y]]$ maximal in the
generic formal

\smallskip

fiber of $B$ but $P$ not in the generic formal fiber of $C$.

\end{slide}

\begin{slide}

\heading{The Generic Fiber of $B \hookrightarrow C$}

{\bf Lemma.} $B = K[[x]][y_1, y_2] \hookrightarrow K[y_1,
y_2][[x]] = C$ is not TGF.

\medskip

{\bf Proof.}  There exists  $\sigma \in K[y_1][[x]]$ that is
transcendental over

\medskip

$K[[x]][y_1]$.  Let $\q = (y_2 - \sigma x)K[y_1, y_2][[x]]$ and
define

\medskip

$\pi: K[y_1, y_2][[x]]\to K[y_1, y_2][[x]]/\q \cong K[y_1][[x]]$.
Thus

\medskip

$\pi(y_2)=\sigma x$. If $h\in \q\cap (K[[x]] [y_1, y_2])$, then
$\exists  \,\, s, t \in \N$ so that

\medskip

$h=\sum_{i=0}^s \sum_{j=0}^t(\sum_{k\in\N} a_{ijk}x^k)y_1^iy_2^j
$, \quad
 where $a_{ijk}\in K$.

 \medskip

Hence  $0=\pi (h)= \sum_{i=0}^s \sum_{j=0}^t(\sum_{k\in\N}
a_{ijk}x^k)y_1^i(\sigma x)^j $.

\medskip

Since $\sigma$ is transcendental over $K[[x]][y_1]$, each
$a_{ijk} = 0$.

\medskip

Therefore  $\q\cap (K[[x]] [y_1, y_2])=(0)$, and $B
\hookrightarrow C$ is not TGF.


\end{slide}





\begin{slide}
{\bf Lemma. } Let $R:=K[[X]]$ and let $P \in \Spec R$ with $x_1 \not\in P$ and 
$\hgt P = r$, where $1 \le  r
\le n-1$. There exists  a change of variables $x_1 \mapsto
z_1:=x_1$ ($x_1$ is fixed), $x_2 \mapsto z_2,\,\hdots, \,x_n
\mapsto z_n$ and a regular sequence  $f_1,\hdots,f_r\in P$ so
that, upon setting $Z_1 = \{z_1, \ldots, z_{n-r}\}$, $Z_2 =
\{z_{n-r+1}, \ldots, z_n\}$ and $Z = Z_1 \cup Z_2$,

 \begin{enumerate}
\item $(f_1, \ldots, f_r) \subset K[[Z_1]][Z_2]$ and $P$ is a
minimal prime of

 $(f_1,\hdots,f_r)R$.

\item  The $(Z_2)$-adic completion  of $K[[Z_1]]\,[Z_2]_{(Z)}$ is
identical to the

$(f_1,\hdots,f_r)$-adic completion
 and is  $R=K[[X]] = K[[Z]]$.

\item  If $P_1 := P \cap K[[Z_1]]\,[Z_2]_{(Z)}$, then $P_1R = P$, that is, 
$P$ is

extended from $K[[Z_1]]\,[Z_2]_{(Z)}$.

\item The extension: $ K[[Z_1]]\hookrightarrow
K[[Z_1]]\,[Z_2]_{(Z)}/P_1\cong K[[Z]]/P$

is finite (and integral).
\end{enumerate}
\end{slide}



\begin{slide}

 \vskip 18 pt

\setbox4=\vbox{\hbox{%
   \rlap{\kern1.7in\lower0in\hbox to .3in{\hss $R =K[[X]] = K[[Z_1, Z_2]]$ \hss}}%
   \rlap{\kern.70in\lower0.8in\hbox to .3in{\hss $(X)R$ \hss}}%
  \rlap{\kern2.70in\lower1.1in\hbox to .3in{\hss $D =K[[Z_1]]\,[Z_2]_{(Z)}$
\hss}}%
    \rlap{\kern.70in\lower1.55in\hbox to .3in{\hss $P=P_1R$ \hss}}%
   \rlap{\kern1.70in\lower2.3in\hbox to .3in{\hss $P_1 =P\cap D$ \hss}}%
   \rlap{\special{pa 1700 80} \special{pa 700 650} \special{fp}}%1
   \rlap{\special{pa 1700 80} \special{pa 2700 950} \special{fp}}%1
       \rlap{\special{pa 700  880} \special{pa 700 1350} \special{fp}}%1
       \rlap{\special{pa 700  1630} \special{pa 1700 2150} \special{fp}}%1
           \rlap{\special{pa 2700  1180} \special{pa 1700 2150}
\special{fp}}%1
    }}

\box4 \vskip 12 pt
%\centerline{The rings of the example} \vskip 18 pt
\medskip
\center{ $(f_1, \ldots, f_r) \subset P_1$ \quad and \quad
$K[[Z_1]] \hookrightarrow D/P_1 \cong K[[Z]]/P$ is finite.}


\end{slide}



\begin{slide}


{\bf Theorem. } Let $y$ and $X=\{x_1,\hdots,x_m\} $ be variables
over the field

$K$. Assume  $V$ is a DVR with $K[y]\subseteq V\subseteq K[[y]]$
such that $V$ has

completion $K[[y]]$. Also assume  the field $K((y))=K[[y]]\,[1/y]$
has

{\bf uncountable} transcendence degree over the fraction  field
${\Q}(V)$ of

$V$. Set $R_0:=V[[X]]$ and $R = \widehat R_0=K[[y,X]]$. Let $P$ be
a prime

ideal of $R$ such that:

 \qquad $P\subseteq (X )R$  (so $y\notin P$)  \quad and \quad  $\dim(R/P) > 2.$

Then there exists  a prime ideal $Q$ of $R$ such that
\begin{enumerate}
\item  $P\subset Q \subset XR, $ \item $\dim (R/Q) = 2$, and \item
$P\cap R_0 = Q\cap R_0.$
\end{enumerate}
In particular, $P\cap K[[X]]= Q\cap K[[X]]$.

\end{slide}

\begin{slide}

\setbox4=\vbox{\hbox{%
   \rlap{\kern1.0in\lower0in\hbox to .3in{\hss $R=K[[y,Z]]$ \hss}}%
    \rlap{\kern.25in\lower.2in\hbox to .3in{\hss \hss}}%
     \rlap{\kern1.0in\lower0.6in\hbox to .3in{\hss
$D:=K[[y,Z_1]]\,[Z_2]_{(Z)}$ \hss}}%
   \rlap{\kern.25in\lower.8in\hbox to .3in{\hss 
\hss}}%
     \rlap{\kern3.0in\lower.4in\hbox to .3in{\hss $\Q(K[[y]])=K[[y]]\,[1/y]=K((y))$ \hss}}%
    \rlap{\kern1.0in\lower1.15in\hbox to .3in{\hss $K[[y]]$ \hss}}%
   \rlap{\kern3.0in\lower1.00in\hbox to .3in{\hss $E:=\Q(T)=L(\gamma_2,\hdots,\gamma_{n-r})$ \hss}}%
    \rlap{\kern1.8in\lower1.6in\hbox to .3in{\hss
$T:=L(\gamma_2,\hdots,\gamma_{n-r})\cap K[[y]]$ \hss}}%
    \rlap{\kern1.8in\lower2.2in\hbox to .3in{\hss $S:=\Q(V(\Delta))\cap
K[[y]]$ \hss}}%
      \rlap{\kern3.0in\lower1.8in\hbox to .3in{\hss $L:=\Q(S)=\Q(V(\Delta))$ \hss}}%
   \rlap{\kern1.73in\lower2.8in\hbox to .3in{\hss $K[y]\subseteq V\phantom{XXXX}$ \hss}}%
  \rlap{\kern3.0in\lower2.3in\hbox to .3in{\hss $\Q(V)$ \hss}}%
           \rlap{\special{pa 1000 80} \special{pa 1000 450} \special{fp}}%1
       \rlap{\special{pa 1000 680} \special{pa 1000 995} \special{fp}}%1
      \rlap{\special{pa 3000 480} \special{pa 1000 995} \special{fp}}%1
        \rlap{\special{pa 3000 480} \special{pa 3000 900} \special{fp}}%1
       \rlap{\special{pa 3000 1050} \special{pa 3000 1680} \special{fp}}%1
         \rlap{\special{pa 3000 1050} \special{pa 1755 1530} \special{fp}}%1
                 \rlap{\special{pa 1800 1650} \special{pa 1800
2080}\special{fp}}%1
                 \rlap{\special{pa 3050 1850} \special{pa 1800
2080}\special{fp}}%1
\rlap{ \special{pa 1050 1200}\special{pa 1800 1530}\special{fp}}%1
                   \rlap{\special{pa 1800  2250} \special{pa 1800
2680}\special{fp}}%1h
           \rlap{\special{pa 3050  1850} \special{pa 3050
2180}\special{fp}}%1
          \rlap{\special{pa 3000  2350} \special{pa 1800
2680}\special{fp}}%1
               }}

\box4 \vskip 12 pt
%\centerline{The rings of the example} \vskip 18 pt


\end{slide}



\begin{slide}
\heading{Prime Ideals With The Same Contraction}

{\bf  Corollary. } Let $K$ be a field, $y$ and
$X=\{x_1,\hdots,x_m\} $ variables

over  $K$ and let $R=K[[y,X]]$. Assume  $P\in \Spec R$ is such
that:

(i) $P\subseteq (x_1,\hdots,x_m)R$
 and

(ii)  $\dim(R/P)
> 2.$

Then there exists  a prime ideal $Q$ of  $R$ such  that

\begin{enumerate}
\item  $P\subset Q \subset XR$, \item  $\dim (R/Q) = 2$, and \item
$P\cap K[y]_{(y)}[[X]] = Q\cap K[y]_{(y)}[[X]]$.
\end{enumerate}
In particular, $P\cap K[[x_1,\hdots,x_m]]= Q\cap
K[[x_1,\hdots,x_m]]$.

\end{slide}

\begin{slide}
\setbox4=\vbox{\hbox{%
   \rlap{\kern1.7in\lower0in\hbox to .3in{\hss $R=K[[y,X]] $ \hss}}%
   \rlap{\kern.7in\lower0.8in\hbox to .3in{\hss $(X)R$ \hss}}%
  \rlap{\kern2.7in\lower1.1in\hbox to .3in{\hss $R_0 = K[y]_{(y)}\,[[X ]]$
\hss}}%
    \rlap{\kern.7in\lower1.55in\hbox to .3in{\hss $P \subset Q$ \hss}}%
   \rlap{\kern1.7in\lower2.3in\hbox to .3in{\hss $P\cap R_0 = Q \cap R_0$ \hss}}%
   \rlap{\special{pa 1700 80} \special{pa 700 650} \special{fp}}%1
   \rlap{\special{pa 1700 80} \special{pa 2700 950} \special{fp}}%1
       \rlap{\special{pa 700  880} \special{pa 700 1350} \special{fp}}%1
       \rlap{\special{pa 700  1630} \special{pa 1700 2150} \special{fp}}%1
           \rlap{\special{pa 2700  1180} \special{pa 1700 2150}
\special{fp}}%1
    }}

\box4 \vskip 12 pt
%\centerline{The rings of the example} \vskip 18 pt

\center{$\dim R/Q = 2$ \quad and \quad $P \cap K[[X]] = Q \cap
K[[X]]$}

\end{slide}



\begin{slide}
\heading{Generic Fibers of Power Series Extensions}

{\bf Theorem. } Let $y$ and $X=\{x_1,\hdots,x_m\} $ be variables
over the field

$K$. Consider the extension  $\widehat A = K[[X]] \hookrightarrow
K[[X]][[y]] = \widehat B$. Let $P$

be a prime ideal of $\widehat B$ such that $P \cap \widehat A =
(0)$. Then
\begin{enumerate}
\item If $P \not\subseteq X\widehat B$, then $\dim \widehat B/P =
m$ and $P$ is maximal with respect

to $P \cap \widehat A = (0)$. \item If $P \subseteq X\widehat B$,
then there exists $Q$ with $P \subseteq Q$, $\dim \widehat B/Q =
2$,

 and $Q$ is maximal with
respect to $Q \cap \widehat A = (0)$.
\end{enumerate}

If $m > 2$, then for each prime $Q$ maximal with respect to

$Q \cap \widehat A = (0)$, we have either
\begin{enumerate}
\item[(i)]  $\dim \widehat B/Q = m$ and $\widehat A
\hookrightarrow \widehat B/Q$ is finite, or \item[(ii)] $\dim
\widehat B/Q = 2$ and $Q \subset X \widehat B$.
\end{enumerate}


\end{slide}



\begin{slide}
\heading{Generic Fibers of Power Series Extensions 2}

{\bf Theorem.} Let $K$ be a field, $m$ and $n$ positive integers,
$X = \{x_1, \ldots, x_m\}$ and  $Y = \{y_1, \ldots, y_n\}$ sets of
independent

variables over $K$.
 Consider the extension

 \medskip

 $$\widehat A = K[[X]]
\hookrightarrow K[[X]][[Y]] = \widehat B.
$$

Let $Q$ be a prime ideal of $\widehat B$ maximal with respect to
$Q \cap \widehat A = (0)$.

1. If $m = 1$, then $\dim \widehat B/Q = 1$ and $\widehat A
\hookrightarrow \widehat B/Q$ is finite.

2. If $m \ge 2$, there are two possibilities:
\begin{enumerate}
\item[(i)] $\widehat A \hookrightarrow \widehat B/Q$ is finite, in
which case $\dim \widehat B/Q = \dim \widehat A = m$, or
\item[(ii)] $\dim \widehat B/Q = 2$.
\end{enumerate}




\end{slide}









\begin{slide}
\heading{Mixed Polynomial/Power Series Rings}

Consider

 $$
 A:=K[x,y]\hookrightarrow B:=K[[x]][y]\hookrightarrow
C:=K[y][[x]]
$$

$$
\hookrightarrow  D:=K[y][[x/y]]\hookrightarrow E:=K[y,1/y][[x]]
$$
\medskip

where $K$ is a field and $x$ and $y$ are indeterminates over $K$.

\medskip

We are interested in   the prime spectra of these rings and the
maps

on the spectra  determined by the inclusion maps on the rings. For

example, do there exist nonzero primes of one of the larger rings

that intersect a smaller ring in zero (i.e. non-TGF extensions).
\end{slide}

\begin{slide}

\heading{Motivation 2}

 From the introduction to a paper  by Alonzo-Tarrio, Jeremias-Lopez
and Lipman:

If a map between  noetherian formal schemes can be factored as a
closed immersion followed by an open one, can  it  also be
factored as an open immersion followed by a closed one?
\medskip

Brian Conrad observed that a counterexample can be constructed for
every triple $(R, x, p)$, where

\begin{enumerate} \item $R$ is an adic domain, that is,    $R$ is a Noetherian
domain that is separated and complete with respect to the powers
of a proper   ideal $I$.

\item
 $x \in R$ is  nonzero,  the completion
of $R[1/x]$ with respect to the powers of $IR[1/x]$, denoted $S: =
R_{\{x\}}$,  is  an integral domain.


\item $p$ is a nonzero prime ideal of $S$ that intersects $R$ in
$(0)$.
\end{enumerate}

\medskip


\end{slide}


\begin{slide}

\heading{Motivation 2 continued}

If  $(R, x, p)$ is such a triple  with $S = R_{\{x\}}$, then the
composition $R \to S \to S/p$ determines a   map on formal spectra
$\text{Spf}(S/p)\to \text{Spf}(S) \to \text{Spf}(R)$ that is a
closed immersion followed by an open one.


For a surjection such as $S \to S/p$ of adic rings gives rise to a
closed immersion $\Spf(S/p) \to \Spf(S)$,  while a localization
like that of $R$ with respect to $x$ followed by the completion of
$R[1/x]$ with respect to the powers of $IR[1/x]$ to obtain $S$
gives rise to an open immersion $\Spf(S) \to \Spf(R)$.

However, the map $\Spf(S/p) \to \Spf(R)$ cannot be factored as an
open immersion followed by a closed one. For a closed immersion
into $\Spf(R)$ corresponds to a surjective map of adic rings $R
\to R/J$, where $J$ is an ideal of $R$.

\bigskip

\end{slide}

\begin{slide}
\heading{Motivation 2 continued}

If the immersion $\Spf(S/p) \to \Spf(R)$ factored as an open
immersion

followed by a closed one, we would have $R$-algebra homomorphisms

from $R \to R/J \to S/p$, where $\Spf(S/p) \to \Spf(R/J)$ is an
open

immersion. Since  $p \cap R = (0)$, we must have $J = (0)$. This
implies

$\Spf(S/p) \to \Spf(R)$ is an open immersion, that is, the
composite

map $\Spf(S/p) \to \Spf(S) \to \Spf(R)$, is an open immersion. But
also

$\Spf(S) \to \Spf(R)$ is an open immersion. It follows that

$\Spf(S/p) \to \Spf(S)$ is both open and closed. Since $S$ is an
integral

 domain this implies $\Spf(S/p) \cong \Spf(S)$. This is a
contradiction

since $p$ is nonzero.

One of our motivations is to describe such triples $(R, x, p)$.


\end{slide}

\begin{slide}

\heading{An example of $(R, x, p)$.}

An example described in [AJL]: Let $w,x,y,z$ be indeterminates

over a field $K$ and consider
$$
R:=K[w,x,z][[y]] \hookrightarrow  S:=R_{\{x\}}= K[w,x,1/x,z][[y]].
$$

Let  $P=(w,z)R$ and $T:=R_P\subset S_{PS}:=G$.



Then  $T\subset G$ are 2-dim  regular local domains, and the
residue

field of $G$ (i.e. the fraction field of $K[x,1/x][[y]]$) is
transcendental

over that of $T$ (i.e. the fraction field of $K[x][[y]]$).


  It follows by [HR, p. 364, Theorem 1.12]
that there exist infinitely

many height-one prime ideals of $G$ in the generic fiber over $T$.
Any

one of these prime ideals of $G$ contracts in $S$ to a prime $p$
as above.


\end{slide}


\begin{slide}

\heading{Is There A Simpler Example?}
\medskip

In some correspondence to Lipman, Conrad asked:

\medskip
{\bf Question.} Is there a nonzero prime ideal of
$E:=K[x,1/x][[y]]$ that intersects $C = K[x][[y]]$ in zero?

\medskip

If there were such a prime ideal, then
$$C:=K[x][[y]] \hookrightarrow
E:=K[x,1/x][[y]]
$$
would be  a simpler counterexample to the assertion that a closed
immersion followed by an open one also has a factorization as an
open immersion followed by  a closed one.

We show there is no such prime ideal.

\end{slide}



\begin{slide}
\heading{A two-dimensional TGF extension.}

Let $K$ be a field and let $x$ and $y$ be indeterminates over $K$.

\medskip


{\bf  Remark.}  $C:=K[x][[y]]\hookrightarrow
E:=K[x,1/x][[y]]$ is 
 a TGF extension.

\medskip

It is true that $\Q(E)$ has infinite transcendence degree
over $\Q(C)$.

Notice that $\dim C = \dim E = 2$ and $y$ is in every maximal
ideal 

of $C$ or $E$.
Let $P$ be a nonzero prime of $E$. If $y \in P$, then

$P \cap C \ne (0)$. Assume $y \not\in P$ and $P \ne (0)$. Then

$K[[y]] \hookrightarrow C/(P \cap C) \hookrightarrow E/P$ and $E/P$
is finite over $K[[y]]$ and 

hence also over $C/(P \cap C)$, so $P \cap C \ne (0)$.

\medskip 

{\bf Question.} Is $K[x, z][[y]] \hookrightarrow K[x, 1/x, z][[y]]$ a TGF extension?






\end{slide}

\begin{slide}
\begin{center}
\textbf{An Example of a TGF extension.}
\end{center}

\bigskip
{\bf Example.} The extension  $R := K[[x,y]]\hookrightarrow S :=
K[[x]][[\frac{y}{x}]]$ is a

TGF extension of the type described in Hochster's question.

\medskip

 To see  this extension is TGF, it suffices to
show $P \cap R \ne (0)$

for each $P \in \Spec S$ with $\hgt P = 1$.

\medskip

  This is clear if $x \in P$, while
if $x \not\in P$, then  $K[[x]] \cap P = (0)$, so

$K[[x]] \hookrightarrow  R/(P \cap R) \hookrightarrow  S/P$ and
$S/P$ is finite over $K[[x]]$.

Therefore $\dim R/(P \cap R) = 1$, so $P \cap R \ne (0)$.

\medskip

{\bf  Remark.} The extension $K[[x, y]] \hookrightarrow K[[x,
\frac{y}{x}]]$ is, up to 

\medskip

isomorphism,  the same as the extension
$K[[x, xy]] \hookrightarrow K[[x, y]]$.

\medskip



\end{slide}




















\begin{slide}


\end{slide}

\begin{slide}


\end{slide}

\begin{slide}


\end{slide}

\end{document}



\begin{slide}

\begin{thebibliography}{99}
\bibitem{BH} W. Bruns and J. Herzog, {\it Cohen-Macaulay
Rings},Cambridge University Press, Cambridge 1993.

\bibitem{ES} P. Eakin and A. Sathaye, {\it Prestable ideals}, J.
Algebra, \textbf{41} (1976) 439-454.

\bibitem{Eb} D. Eisenbud, {\it Commutative Algebra With a View Towards
Algebraic Geomety}, New York, 1995.

\bibitem{EG} D. Eisenbud and S. Goto, {\it Linear free resolutions and minimal multiplicity},
J. Algebra. \textbf{88} (1984), 89--133.

\bibitem{JE} J. Elias, {\it Characterization of the Hilbert-Samuel
Polynomials of Curve Singularities}, Composito Mathematica
\textbf{74} (1990) 135-155.

\bibitem{HW} J. Herzog and R. Waldi, {\it A Note on the Hilbert
Function of a One-Dimensional Cohen-Macaulay Ring}, Manuscripta
Math. \textbf{16} (1975) 251-260.

\bibitem{DK} D. Kirby, {\it The Reduction Number of a
One-Dimensional Local Ring.} J.London Math. Soc.
\textbf{10}(1975), 471-481.

\bibitem{Marley} T. Marley, {\it The Reduction Number of an Ideal
and the Local Cohomology of the Associated Graded Ring.}  Proc.
Amer. Math. Soc. \textbf{117} (1993), 335-341.

\bibitem{M} H. Matsumura, {\it Commutative Ring Theory}, Cambridge
University Press, Cambridge 1986.

\bibitem{RTV1} M. Rossi, N.V. Trung, and G. Valla, {\it
Castelnuovo-Mumford Regularity and Finiteness of the Hilbert
Functions}, Preprint, 2004.

\bibitem{RTV2} M. Rossi, N.V. Trung, and G. Valla, {\it
Castelnuovo-Mumford Regularity and Extended Degree} Trans. Amer.
Math. Soc. \textbf{355}, N.5 (2003), 1773-1786.

\bibitem{Tr} N.V. Trung {\it Reduction Exponent and Degree Bound
for the Defining Equations of Graded Rings}, Proc. Amer. Math.
Soc. \textbf{101} (1987), 229-236.

\bibitem{Tr2} N.V. Trung {\it The Castelnuovo Regularity of the
Rees Algebra and the Associated Graded Ring}, Transactions of the
American Mathematical Society. \textbf{350} (1998), 2813-2832.

\bibitem{VV} P. Valabrega and G. Valla, {\it Form Rings and Regular
Sequences}, Nagoya Math J \textbf{72} (1978)  93-101.

\bibitem{Valla} G. Valla {\it Problems and results on Hilbert functions
of graded algebras.}, Six lectures on commutative algebra.
Bellaterra, 1996. 293-344.

\end{thebibliography}

\end{slide}

\end{document}
